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\A NUMERICAL SIMULATION OF TWO-DIMENSIONAL
SEPARATED FLOW IN A SYMMETRIC
OPEN-CHANNEL EXPANSION USING THE DEPTH-INTEGRATED TWO-EQUATION
(k-e) TURBULENCE CLOSURE MODEL;
by
Raymond Scott ,,Chapman,,
Dissertation submitted to the Graduate Faculty of the
Virginia Polytechnic Institute and State University
in partial fulfillment of the requirements for the degree of
DOCTOR OF PHILOSOPHY
In
Civil Engineering
APPROVED:
C. Y. Ku4, Chairman
C. H. Lewis H. W. Tieleman
, B.
May,
Blacksburg, Virginia
ACKNOWLEDGEMENTS
The author extends his sincerest thanks to the members of his ad-
visory committee, Dr. Chin Y. Kuo, Dr. Clark H. Lewis, Dr. Henry W.
Tieleman, Dr. James M. Wiggert, and (from the USAE Waterways Experiment
Station, Vicksburg, Mississippi) Dr. Billy H. Johnson.
A significant portion of the present study was carried out while
the author was on contract to the Environmental Laboratory of the USAE
Waterways Experiment Station through an Intergovernmental Personnel Act
Assignment Agreement with Virginia Polytechnic Institute and State Uni-
versity. Of the many people within WES that have provided valuable as-
sistance, the author would particularly like to acknowledge the efforts
of Mr. Ross W. Hall, Jr., Principal Investigator, Marsh Estuarine Model-
ing Team, and Mr. Donald L. Robey, Chief, Ecosystem Research and Simula-
tion Division, Environmental Laboratory.
Finally, the author acknowledges the support of the National
Science Foundation, grant No. CME-8004364, which made the initial phases
of this study possible.
ii
TABLE OF CONTENTS
ACKNOWLEDGMENT ii
TABLE OF CONTENTS iii
CHAPTER I INTRODUCTION 1
CHAPTER II DEPTH INTEGRATION OF THE EQUATIONS OF MOTION 11
CHAPTER III CLOSURE MODELS 18
CHAPTER IV COMPUTATIONAL METHOD 35
CHAPTER V MODEL SIMULATIONS 45
CHAPTER VI RESULTS AND DISCUSSION 53
CHAPTER VII SUMMARY AND CONCLUSIONS 62
REFERENCES 66
NOMENCLATURE 72
ILLUSTRATIONS 76
VITA 106
ABSTRACT
iii
CHAPTER 1
INTRODUCTION
In recent years, the field of computational hydraulics has matured
at a remarkable rate of development. The reason for this rapid evolu-
tion is essentially two fold. Firstly, many of the problems encountered
by hydraulic engineers, such as heat and mass transport, fluvial geo-
morphology, fluid structure interaction, and wave propagation, can not
be adequately analyzed using close form linear or approximate solutions
to the governing equations of motion. Secondly, the advent of larger
and faster computers has made the use of sophisticated numerical tech-
niques almost commonplace.
The role of the computer, and its impact on hydraulics in general,
is uniquely characterized in the first paragraph of the pref ace of
Computational Hydraulics (Abbott 1979) which reads:
Very much as the steam engine was the principal physical instru-ment of the industrial revolution, so the digital computer is the principal physical instrument of our current "informational revo-lution". The digital computer's capacity to transform semantic information rapidly, reliably and cheaply, changes all technolog-ies, even one with such long historical traditions as hydraulics. One of the first studies of the European enlightenment, investi-gated by da Vinci, Galileo, Newton and Euler, hydraulics had already been transformed through the industrial revolution into a 'productive benefit'. The current post-industrial revolution, concerned with directing investment into complex, finely balanced, 'high-information' machines and constructions, maintaining 'the highest information level per rate of information destruction', transform hydraulics again, adapting it to the new demands and the new possibilities of our modern societies.
One aspect of hydraulics that has been rather successful in adapt-
ing itself to modern technological needs is that area of computational
1
2
hydraulics that addresses the theory and computation of free surface
flows. Much of this success is attributable to the fact that a wide
range of engineering problems, which are complicated by free surface
interaction, have been overcome by the introduction of the depth inte-
grated equations of motion (Hansen 1962).
For this class of nearly horizontal free surface flow problems,
the three-dimensional, time-averaged equations of fluid motion may be
integrated over the vertical dimension thereby reducing the dynamical
problem to solving for the horizontal depth-integrated velocities, and
the water surface elevation. A consequence of spatially averaging the
equations of motion is that additional closure approximations are re-
quired to represent the resulting bottom stress, wind stress, and the
so called effective stresses. The effective stresses, as defined by
Kuipers and Vreugdenhil (1973) consist of the depth-integrated viscous
stresses, turbulent Reynold's stresses, and additional stresses which
result from depth averaging the nonlinear convective acceleration terms
(here after called momentum dispersion).
In an attempt to improve upon the current "state of the art" of
free surface simulation models, the goal of the present study is to
develop a two-dimensional depth-integrated hydrodynamic model capable of
accurately predicting local regions of recirculation induced by abrupt
changes in channel geometry. To accomplish this goal, it will be neces-
sary to 1) implement a sufficiently accurate finite difference technique
to minimize spurious numerical effects, and 2) employ a reasonable clo-
sure approximation for the effective stresses. The overall performance
3
of the simulation model will be assessed in a comparison of model
predictions of steady, free surface, separated flow in a wide, shallow,
rectangular channel with an abrupt expansion in width, with experimental
measurements.
Literature Review
Experimental Measurement of Channel Expansion Flows
The first comprehensive study of separated, turbulent flow in a
channel expansion was performed by Abbott and Kline (1962). Using a
dye flow visualization technique Abbott and Kline examined the effects
of the area ratio (AR = 1 + w1/W0 ) on the nondimensional reattachment
length, xr/W1 , for flow past single and double backward facing steps in
a water channel (Figure 1). For area ratios greater than 1.5, Abbott
and Kline showed that the expansion flow was asymmetric with the reat-
tachment length on one side of the double step configuration being about
three times that on the other. However, for area ratios less than 1.5,
the reattachment lengths were symmetric and equal to a single step reat-
tachment length (x /W1 = 7 + 1) of the same area ratio. r -
Since the early work of Abbott and Kline (1962), a number of simi-
lar experimental studies of flow over backward facing steps have
appeared in the literature. A review of these works may be found in
Kim et al. (1978), Durst and Tropea (1981), and Eaton and Johnston
(1981). An important point brought out by Durst and Tropea (1981), and
Eaton and Johnston (1981) was that observed reattachment length varied
considerably from one experimental investigation to the next. For
4
example, in a compilation of data for twelve independent experimental
studies of symmetric channel expansion flow, Durst and Tropea (1981)
show a variation in the observed nondimensional reattachment lengths of
4.5 ~ xr/W1 ~ 8.
It has been suggested (Durst and Tropea 1981, Eaton and Johnston
1981) that much of the variation in the observed reattachment length is
attributable to differences in 1) the area ratio, AR, of the expansion
section and 2) the aspect ratio of the flow apparatus, h/W1 , where h
is defined to be either the fluid depth, or the duct height. For
symmetric channel expansion (AR< 1.5), Abbott and Kline (1962) showed
the reattachment length to be reasonably constant, which suggests that
the aspect ratio must be of greater importance for this class of expan-
sion flows.
The effect of the aspect ratio of the channel was in fact studied
by de Brederode and Bradshaw (1972) who discovered that the reattach-
ment length was essentially unaffected if the aspect ratio was greater
than about ten. However, they found that for a fully turbulent flow at
the point of separation, the reattachment length decreased for aspect
ratios less than ten.
The observation of de Brederode and Bradshaw may be further re-
fined by simply plotting published measurements of reattachment lengths
for symmetric channel expansions (single and double step) as a function
of an aspect ratio, h/W , based on the inlet channel half-width, W , 0 0
(Figure 2). The trend of decreasing reattachment length with decreasing
aspect ratio is clearly illustrated in figure 3.
5
In addition to reattachment length measurements, detailed measure-
ments of the velocity distribution within the recirculation region of
symmetric expansions are presented by Etheridge and Kemp (1978), and
Moss et al. (1977).
Effective Stress Concept
The first examination of the significance of the effective stresses
was in a study on the generation of secondary flows presented by Kuipers
and Vreugdenhil (1973). The basis of their theoretical analysis was the
derivation of the depth-averaged vorticity equation. Conclusions drawn,
that are of interest in the present work, were (1) vorticity is gen-
erated by the convergence and divergence of the depth mean velocity
field, (2) the bottom stress acts to dissipate vorticity, and (3) the
net moments of the effective stresses relative to vertical axis can gen-
erate vorticity in either direction.
It is interesting to note that in their secondary flow simula-
tions, Kuipers and Vreugdenhil chose to neglect the effective stress
terms; however, they do show that the effective stresses were repre-
sented in a fashion by the spatial smoothing procedure used to overcome
nonlinear instability. Based on a comparison of predicted model results
with hydraulic model studies, Kuipers and Vreugdenhil suggest: (1) sec-
ondary flow circulation can be predicted by a depth-integrated simula-
tion model, (2) the convective acceleration terms must be retained,
(3) horizontal eddies are not reproduced if less than five computational
grid points cover its diameter, (4) the generation of eddies is sensitive
6
to the magnitude of the bottom friction coefficient, and (5) care must be
taken in the treatment of wall boundaries.
The conclusions of Kuipers and Vreugdenhil have recently been
verified in a paper by Ponce and Yabusaki (1981). In their study Ponce
and Yabusaki adopted the work of Kuipers and Vreugdenhil and applied it
to flow past a square cavity and a channel expansion.
Flokstra (1976, 1977) extended the theoretical work of Kuipers and
Vreugdenhil (1973) by integrating the depth averaged vorticity equation,
over a region of steady flow bounded by a closed streamline, to show
that the dissipation of vorticity due to bottom friction is balanced
exactly by the generation of vorticity by the effective shear stresses.
As a result, he concluded that model simulations of main flow driven
recirculation, performed with the effective stress neglected, show only
the effects of numerical errors on the solution. By means of an energy
balance, Flokstra further showed that momentum dispersion generally
transfers energy out of the region of circulating flow leaving the tur-
bulent Reynold's stresses to act as the mechanism which transforms energy
into the region. Consequently, he concluded that with respect the effec-
tive stress closure, attention has to be concentrated mostly on modeling
the effects of the depth-integrated Reynold's stresses.
Turbulent Reynold's Stress Closure
In the recent literature, the treatment of Reynolds stress closure
has ranged from neglecting the terms to the concept of "large eddy"
simulations (Leonard 1974) where, by means of spatial filtering of the
equations of motions, only the small scale or "subgrid scale" Reynolds
7
stresses need be modeled. Within these two extremes lie a number of
alternative closure schemes which exhibit a wide variation in complexity
(Reynolds 1976).
One closure technique that has enjoyed considerable success in the
simulation of a variety of turbulent flows is the two equation (k-e)
turbulence model described by Launder and Spalding (1974) and Rodi
(1980). Applications. of a depth-integrated version of the (k-e) turbu-
lence model have been presented by Rastogi and Rodi (1978), and McQuirk
and Rodi (1978).
Rastogi and Rodi (1978) adopted the parabolized form of the depth-
intergated equations of motion and (k-e) turbulence model to simulate
the problem of a coaxial jet issuing into a rectangular channel. By
restricting their attention to a normal flow configuration with constant
width, they were able to utilize the existing computational scheme of
Patankar and Spalding (1972).
McQuirk and Rodi (1978) extended the use of the depth-averaged
(k-e) model to recirculating flows by simulating a side discharge, slot
jet issuing normally into a rectangular channel. Forcing the free sur-
face boundary to act as a "rigid lid" enabled them to apply the compu-
tation scheme of Gosman and Pun (1973), which was originally designed
for the simulation of confined flows.
Although the results of the two model simulations appeared to com-
pare well with experimental measurements, the general applicability of
the models is limited by two significant problems. First, the specifi-
cation of a "rigid lid" free surface boundary at best allows linear
8
variation in the water surface elevation which may not be adequate when
modeling main flow driven recirculation. Secondly, the computational
schemes adopted in both simulations are based on upwind differencing for
the convective accelerations. Consequently, the simulation results could,
in fact, have been tuned to experimental measurements, via numerical
diffusion, by simply adjusting the nonuniform grid spacing (Castro 1977).
Momentum Dispersion Closure
The development of closure schemes for the momentum dispersion
mechanism (Flokstra 1977, Abbott and Rasmussen 1977, Lean and Weare
1979) has, to date, solely been based on a simple helicoidal flow approx-
imation, which allows the vertical variation of the horizontal velocity
components to be modeled by known theoretical velocity distributions
(Van Bendegom 1947, Rozovskii 1957, Engelund 1974). Although the theo-
retical velocity distributions used to obtain closure models for
momentum dispersion were all derived under the assumption of fully de-
veloped flow in a long, channel bend, and subsequently do not strictly
apply to flow through channel expansion; important information regarding
the nature of the momentum dispersion mechanism can be gotten from these
models. First, the net effect of the normal components of momentum dis-
persion is to increase the effective shear stress, irrespective of the
degree of streamline curvature. Second, the magnitude of the transverse
component of momentum dispersion, which can act to decrease the effective
shear stress, is directly proportional to the ratio of the depth of flow
to the radius of curvature of the depth-mean streamlines.
Nonetheless, any attempt at incorporating an illfounded closure
9
model for momentum dispersion, in the present study, would only serve
to complicate the assessment of turbulence closure. Consequently, the
direct representation of momentum dispersion mechanism will not be
addressed herein, however, the indirect effect of momentum dispersion on
turbulence closure will be discussed in Chapter 3.
Computational Method
Recently, higher order finite difference techniques have been suc-
cessfully applied to strongly convective transport problems in both one
and two dimensions (Abbott and Rassmussen 1977, Hinstup et al. 1977,
Holly and Preissmam 1977, Leonard 1979, Leschziner 1980, Chapman and Kuo
1981, Leschziner and Rodi 1981, Han et al. 1981). One of the more nota-
ble techniques for steady flow problems is the QUICK (Quadratic Upstream
Interpolation for Convective Kinematics) method developed by Leonard
(1979). The QUICK finite difference technique, which is based on a con-
servation, control volume integral formulation possesses the desirable
convective stability of upwind differencing, but is free of what is
classically called numerical diffusion (Roache 1972).
Comparison of QUICK and upwind differencing shows QUICK to be far
superior in turbulent transport simulations where practical grid spacing
were used (Leonard et al. 1978, and Leschziner and Rodi 1981). It is
interesting to note that both Leonard et al. 1978) and Leschziner and
Rodi 1981), point out that when the standard (k-e) turbulence model was
modified to reflect the influence of streamline curvature (Bradshaw 1968,
Launder et al. 1977, Militzer et al. 1977), the upwind difference solu-
tion exhibited no detectable response to closure modification, where as
10
a marked improvement was seen in the QUICK solution. The conclusion
drawn in these studies was that the numerical diffusion introduced by
upwind differencing simply overwhelmed the effects of streamline curva-
ture modification.
Study Objective
The objective of the present study is to develop a two-dimensional,
depth-integrated free surface hydrodynamic model capable of accurately
simulating local regions of recirculation induced by abrupt changes in
channel geometry. In order to accomplish this objective, the research
program is divided into two phases:
(I) Development and testing of a depth-integrated hydrodynamic
model, based on the QUICK finite difference technique, with effective
stress closure neglected.
(II) A test application of the depth-integrated two-equation (k-e)
turbulence closure model for separated, free surface flow in a wide,
shallow rectangular channel with an abrupt expansion in width.
The performance of the depth-integrated (k-e) turbulence closure
is evaluated by comparison of model simulations with published experi-
mental data.
CHAPTER 2
DEPTH INTEGRATION OF THE EQUATIONS OF MOTION
Adopting tensor notation, the three dimensional, time-averaged
equations of motion for an incompressible, homogeneous fluid with Cori-
olis forces neglected are written as follows:
where
Conservation of Mass
Conservation of Momentum
av. a(v.v.) ____.!. + ]. J at ax.
J
av. ____.!. = 0 ax.
].
1 aP aa .. = -- ~- + g + __..!.J. p ax. i ax. ]. J
v. = time averaged velocity components (u, v, w) ].
x. = Cartesian coordinate directions (x, y, z) ].
t = time
i,j = 1, 2, 3 and repeated indices require summation
p = fluid density
P = pressure
g. = acceleration due to gravity (O, O, -g) ].
a .. = shear stress tensor per unit mass l.J
(1-1)
(1-2)
The shear stress tensor per unit mass a .. l.J
which includes both molecu-
lar and turbulent effects is written in the usual way:
11
where
a .. l.J
12
= -v!v~ + ]. J
( av. av·) v -2:. + _J_ ax. ax. J ].
v! = turbulent velocity fluctuations ].
v = kinematic viscosity
(1-3)
The right-handed orthogonal Cartesian coordinate system used in
the present work is oriented such that the longitudinal direction, x
is positive to the right and the vertical direction, z is positive
upward. For puposes of illustration, a definition sketch is presented
in figure 4.
Depth-Integration Process
For nearly horizontal flows with small depth to width ratios, it
is appropriate to spatially average the equations of motion over the
vertical dimension, z , thereby obtaining a depth-integrated represen-
tation of the flow field. As an example, consider the depth integra-
tion of the x-momentum equation presented by Kuipers and Vreugdenhil
(1973). Defining zb as the channel bottom elevation above some
arbitrary datum, and h as the water depth, Leibnitz's rule is applied
to
h+z [ b au a 2 a I - + - (u) + - (uv) at ax ay zb
+ £.__ (uw) + .! aP az p ax
a (a ) a (crxy) ] dz = 0 ax xx - ay
which when integrated becomes:
+
u dz
13
2 a Jh+zb u dz + ay
z b
UV dz
[ ua ah + u2 ~ (h + zb) + u v ~ (h + zb) - uawa] at a ax a a ay (I)
[ 0 xxa a (h + zb) ax
a crxx dz - ay
1 a p ax
a +a ay (h + zb) xya (III)
(J xy dz
- a J xza
( 0 xxb azb - -+ ax
azb a --xyb ay (IV)
(1-4)
0 xzb) = 0
where the subscripts a and b denote the free surf ace and channel
bottom, respectively.
Recognizing that the terms labeled (I) and (II) correspond ex-
actly to kinematic boundary conditions for the free surface and channel
bottom, respectively, they must vanish. The term labe~ed (III) is the
component of the wind shear stress tensor per unit mass acting along
the free surface. The effect of wind forcing is not essential to the
problem at hand and will subsequently be dropped from further discussion.
14
The term labeled (IV) is the component of the bottom shear stress tensor
per unit mass acting in the plane of the channel bottom. With the
assumption of a nearly horizontal free surface and channel bottom we can
approximate term (IV) as a bulk bottom shear stress per unit mass, tbx .
The parameterization of tbx will be discussed in Chapter 3.
The pressure terms in equation (1-4) can be handled in a consist-
ent fashion by requiring that vertical accelerations are small com-
pared to gravity. Hence the z-momentum equation reduces to:
aP az - -pg (1-5)
Integrating and applying the boundary condition that the pressure equal
atmosphere pressure at the free surface one obtains:
P = pg(h + zb - z) + Pa
where P denotes atmospheric pressure. At this point, it will be a
assumed that atmospheric pressure is both uniform and equal to zero
gage pressure. With the resulting hydrostatic approximation, the sim-
plification of the remaining pressure terms
1 a p ax
will proceed as follows:
(1-6)
15
1 a rb P dz a 1'h (h + zb. - z) dz p ax = g ax zb zb
(1-7) a (h2) = g ax 2
and similarly,
(1-8)
Depth integration of the velocity components in the temporal and
convective acceleration terms is accomplished by redefining the depth
varying horizontal components, u and v , in terms of depth average
values U and V and depth varying deviations, u and ; • The
depth average values are defined by
u dz (1-9)
and
h+zb V = -h1 J v dz
zb
such that
; dz = 0 (1-10)
Following these rules the depth-averaged temporal and convective accel-
eration terms are rewritten:
16
a Jzb u dz a (Uh) (1-11) at = at b
h+zb + l1_ rb :x J 2 dz a (U2h) 2 (1-12) u = ax (u - U) dz ax
zb zb
;~ h+zb a
UV dz a (UVh) + :y J (u - U) (v - V) dz (1-13) ay = ay
b zb
The second term on the right-hand side of equations (1-12) and (1-13)
are momentum dispersion mechanisms which result from representing a non-
uniform vertical boundary layer by a depth-averaged value of the veloc-
ity components.
Realizing the obvious symmetry in the depth integration process,
the conservation of mass and y-momentum equations may be similarly
integrated yielding the complete depth-integrated equations of motion,
which when written in indicial form read:
Conservation of Mass
Conservation of Momentum
acv h) m _a_t_ +
in which
acv v h) m n ax n
acv h) oh + m = 0 at ax m
(1-14)
(1-15)
17
m,n = 1, 2 and repeated indices require summation;
v = two-dimensional depth-averaged velocity vector (U, V) m x = coordinate directions (x, y) m
tbm = components of the bottom shear stress per unit mass
T = components of the depth-integrated effective stress tensor mn per unit mass
The depth-integrated effective shear stress tensor as defined
by Kuipers and Vreugdenhil (1973) and Flokstra (1977) contains the
viscous stresses, the turbulent Reynold's stresses, and the momentum
dispersion terms which arise from depth-integrating the nonlinear con-
vective acceleration terms in the equations of motion. Specifically,
the depth-integrated effective stress tensor per unit mass is written:
' y·b Vi .....
[ (~ av) T - v ax: + ax: -v'v' - (v - V )(v mn m n m m n
zb
in which
v' = horizontal turbulent velocity fluctuations m
- Vn)] dz (1-16)
The closure approximations adopted in the present work to repre-
sent the bottom shear stress, and effective stress mechanisms is the
subject of the next chapter.
CHAPTER 3
CLOSURE MODELS
As in any turbulent flow simulation, a closure problem exists due
to the presence of more unknowns than governing equations. The closure
problem associated with the use of the depth-integrated equations of
motion, in the present study, requires the parameterization of: (1) the
bottom shear stress, (2) the depth-averaged turbulent Reynold's
stresses, and (3) the momentum dispersion terms. In this chapter, the
closure models adopted to represent these mechanisms will be discussed.
Bottom Shear Stress Closure
The bottom shear stress per unit mass is parameterized in accor-
dance with the well-known quadratic shear stress law (Schlichting 1955),
namely
(3-1)
where
tb = magnitude of the resultant bottom shear stress vector per unit mass (tbx' thy)
c = drag coefficient for water flowing over a fixed surface
q = magnitude of the resultant velocity vector (U, V)
Defining theta (B) to be the angle that lies between the resultant
velocity vector and the x-axis, the vector components of the resultant
shear stress per unit mass are written:
18
and
Noting that
and
19
2 = cq cos e
2 t = cq sin e by
u = q cos e
v = q sin e
(3-2)
(3-3)
the familiar form of the bottom stress closure model is written
where
tb = cV q m m
1/2 q = (U2 + V2)
(3-4)
(3-5)
The value of the drag coefficient can be related to the channel
roughness, z , and depth, h , by assuming that the vertical boundary 0
layer in the central portion of an unidirectional, fully developed
20
turbulent open channel flow is adequately described by the logarithmic
velocity law:
U.,. ( ) u(z) = K,.. .£n ~o (3-6)
where
u~ = shear velocity, (t ) 112 ·· bx
K = von Karman's constant
By definition,
(3-7)
where again U is the depth-averaged velocity, and thus
(3-8)
where z is the distance from the channel bottom where u(z) = U .
The value of z is obtained by first recalling that
1 u = (h - z )
0 or
h
f u(z) dz 0
(3-9)
(3-10)
21
Then combining equation (3-10) and equation (3-6) and simplifying yields
(3-11)
where
e = exponential function
Substitution of equation (3-11) into equation (3-8) yields an exact
functional relationship for c in terms of channel roughness and depth,
specifically:
c = (3-12)
It should be noted that equation (3-12) may be applied to flows without
a fully developed boundary layer by simply replacing the depth, h , by
an approximate boundary layer thickness.
Depth-Averaged Turbulent Reynold's Stress Closure
The turbulence closure model used herein is a modification of the
depth-integrated (k - e) presented by Rastogi and Rodi (1978). The
(k - e) turbulence model is based on the Boussinesq eddy viscosity hypo-
thesis (Hinze 1975) which assumes that the turbulent Reynold's stresses
22
are proportional to the mean strain rates. Using three-dimensional
tensor notation, the turbulent Reynold's stresses are written:
where
-v!v~ = l. J
2 - -3ko .. l.J
k = -21 v!v! , the turbulent kinetic energy per unit mass l. l.
o .. = Kronecker delta l.J
(3-13)
Unlike the molecular viscosity, v , the turbulent eddy viscosity is
flow dependent and can vary both in space and time. An approximation
for the distribution of the turbulent eddy viscosity is obtained by
assuming that it is proportional to the product of the characteristic
velocity and length scale of turbulence, namely:
where
(3-14)
Q = the macroscale of turbulence (a measure of the size of the energy containing eddies).
An inviscid estimate of the energy dissipation rate per unit mass, e ,
is obtained when one assumes that the amount of energy dissipated at the
small scales of turbulence equals the rate of supply at the large scales.
23
Again utilizing the characteristic velocity and length-scales of tur-
bulence, dimensional considerations require that (Tennekes and Lumley
1972):
(3-15)
Substitution of equation (3-15) into equation (3-14) yields a functional
relationship for the turbulent eddy viscosity in terms of the kinetic
energy of turbulence, k , and its rate'of dissipation, e ,
specifically:
(3-16)
where CV is an empirical coefficient.
In principal, solution of the exact transport equations for the
turbulence kinetic energy, k , and its rate of dissipation, e , en-
ables one to completely specify the temporal and spatial distribution of
the turbulent eddy viscosity. However, construction of the transport
equations for k and e result in additional closure problem. Con-
sider first the turbulence kinetic energy equation (Hinze 1975):
ak + at
a(kvi) = ~ [v'(~ + ax. ax. i p
1 1 (I)
24
( ) av. v'.v! _J_ 1 J ax.
1 (II)
(3-17)
(av'. av! av!
v _1 + __J_ _J_ ax. ax. ax.
J 1 1 (IV)
where P' denotes turbulent pressure fluctuations, and the over-bar
represents a time average. The closure problem results from the pres-
ence of the unknown pressure and velocity fluctuation correlations in
term (I). Physically, term (I) represents the convective diffusion of
the total turbulence mechanical energy per unit mass by turbulence.
This term acts as a redistribution mechanism which suggests the use of
a gradient diffusion model (Rodi 1980), or
( P' ) _ Vt ak v 1'. -p + k - - --
ak axi (3-18)
where ak is an empirical constant. Equation (3-13) may be substituted
directly for the turbulent Reynold's stresses in the turbulence produc-
tion term (II). Term (IV) is by definition the energy dissipation rate
per unit mass, e . Finally, term (III) represents the work done by the
viscous shear stresses. For high Reynold's number flows, this term is
small and can be neglected (Hanjalic and Launder 1972). Making the ap-
propriate substitutions, the three-dimensional model equation for the
turbulence kinetic energy is written:
25
av. 1
-- - E. ax. J
(3-19)
The exact transport equation for the energy dissipation rate per
unit mass, for large Reynold's number, reads (Harlow and Nakayama 1968):
-2v
where
ae. + a c ) at ax. E.Vi 1
( av'. av'. av') 1 1 r ax ax ax r s s
(II)
- 2
av. 1 = -2v -ax r
(v a2vi_ )2 ax ax r s
(III)
(av'. av' av' __ 1 ____.!'. + s ax ax ax. s s 1
(I)
av~) ax r
a (v'.e.') v a - ax. J - p ax. ( aP' avi_) ax ax s s J 1
(IV) (V)
r,s = 1, 2, 3 and repeated indices require summation
(3-20)
e.' = turbulent fluctuations of the energy dissipation rate per unit mass.
The closure approximations for equation (3-20) were first presented by
Hanjalic and Launder (1972). Their approach was to parameterize (I)-
(III) in terms of the Reynold's stresses, mean strain rate, turbulence
kinetic energy, and its rate of dissipation per unit mass, and neglect
term (V) on the basis of being small. Term (I) represents the produc-
tion mechanism for e. and was approximated accordingly,
26
( av.
(I) = -cl f) (viv~) ax: (3-21)
Terms (II) and (III) were grouped together and parameterized as follows:
(II) + (III) (3-22)
The argument used to support equation (3-22) was that the sum of
term (II), which represents the generation rate of vorticity fluctua-
tions due to the self~stretching mechanism, and term (III), which repre-
sents the viscous decay of dissipation, should be controlled by the
dynamics of the energy cascade. Subsequently, if the Reynold's number
is sufficiently large to allow the existence of an inertial subrange an
inviscid estimate based on dimensional considerations should be appro-
priate. Term (IV) represents the turbulent diffusion of e which
clearly suggests a closure of the form:
(IV) (3-23)
where is an empirical constant. Collecting the various approxima-
tions yields the complete model equation for the energy dissipation rate
per unit mass:
a(v.e) ae + i at ax.
l.
a (v t a ) (av. av·) av. c2e2 = ~- -- ~ + c v ~ ~-l. + __J_ ~-l. -ax. a ax. 1 t k ax. ax. ax. -k-J e J J i J
(3-24)
27
Estimates for the empirical constants found in equations (3-19)
and (3-24) were obtained by applying the model equations to simple tur-
bulent flows for which experimental data were available. For example,
in a local equilibrium, two-dimensional, boundary layer the production
of turbulence energy is balanced by dissipation and Equation (3-19)
reduces to:
--au u'v' - = e ay
A consistent closure approximation for the Reynold's stress is:
which yields --2 u'v'
Now by definition,
(3-25)
(3-26)
(3-27)
(3-28)
thus, substitution of equation (3-27) into equation (3-26) results in
1/2 u'v' c = v -k- (3-29)
Experimental data (Ng and Spalding 1971) extracted from the work of
Laufer (1954) suggests that u'v'/k varies from 0.22 to 0.3 which
28
gives a range of CV from 0.05 to 0.09. The value of c2 was found to
to lie between 1.9 and 2.0 (Hanjalic and Launder 1972, and Launder et al.
1975) when computed using measured decay rates of the turbulent kinetic
energy behind a grid (Batchelor and Townsend 1948, and Townsend 1956).
The constant c1 was obtained by examining the form of the dissipation
equation in the constant shear stress region near the wall. Here con-
vection is negligible with production and dissipation approximately in
balance, thus
u; (3-30) e = Ky
Substituting equation (3-30) into equation (3-24) with some simplifica-
tion yields
(3-31)
In the near wall region (Townsend, 1956)
(3-32)
thus
(3-33)
where by definition
29
and therefore
c1/2 ae v
(3-34)
(3-35)
which specifies the value of c1 when Cv , c2 , ak and ae are
known. The constants ak and ae , which are similar to turbulent
Prandtl or Schmidt Numbers, were assumed to be of order unity and de-
termined along with the recommended values of the other constants via
computer optimization. This was done by adjusting the values of the
various constants until reasonable agreement between computed and ex-
perimental results were obtained. The values recommended by Launder
and Spalding (1974) are as follows:
CV = 0.09
(3-36)
ae = 1.3
30
With the appropriate specifications of boundary and initial conditions,
equations (3-16), (3-19), (3-24), and (3-36) constitute the complete
three-dimensional (k - e) turbulence closure model.
To be of use in approximating the depth-averaged Reynold's stress
in equation (3-16) it is necessary to cast the three-dimensional (k - e)
model into a depth-integrated form. Realizing that turbulence is inher-
ently three dimensional, depth integration of the transport equations
for the turbulence kinetic energy and its rate of dissipation cannot be
strictly performed. However, Rastogi and Rodi (1978) suggest model
equations for the depth-averaged turbulence energy, k , and its rate of
dissipation, e , can, in fact, be constructed if additional source
terms are added to account for mechanisms originating from·the nonuni-
formity of the flow over the vertical dimension. Furthermore, they
suggest that the resulting turbulent viscosity, Vt , should be inter-
preted such that when multiplied times the depth-averaged strain rate
will yield the depth-averaged turbulent Reynold's stress. By analogy,
the depth-averaged Reynold's stress tensor is written
where
"2 0 = c !-t v e
2 ,.. - - kho 3 mn
The resulting model equations for the depth-averaged value of the
(3-37)
(3-38)
31
turbulence energy, k , and its rate of dissipation, e , are written as
follows:
and
where
a(kh) at
a(eh) + at
....
avkh [ .... ] + ( m ) = ~ 0 a(kh) + P ax ax t ax h m m m + Pk - eh (3-39)
(3-40)
(3-41)
The source terms Pk and Pe account for the production mechanism
resulting from the presence of a vertical boundary layer. The form of
these production terms may be obtained by considering the central por-
tion of a unidirectional, uniform flow in a wide open channel. For this
flow, the balance equations for k and € reduce to
and
P - eh k - (3-42)
(3-43)
To a good approximation, the total turbulence energy production over the
vertical is written (Townsend 1956):
but by definition,
therefore,
32
2 pk = u .... u
" (3-44)
(3-45)
P = cu3 (3-46) k
A similar relation may be obtained for the dissipation source Pe by
recalling that from equation (3-38)
which may be substituted into equation (3-43) to yield
Noting that
c c1/2 ,.,3;2h 2 v e
and introducing the nondimensional dispersion coefficient
(3-47)
(3-48)
(3-49)
(3-50)
equation (3-48) is rewritten:
33
C c1/2 5/4u4 2 v c
(3-51)
Generalizing these results to two dimensions is simply a matter of re-
placing the unidirectional flow velocity U with the magnitude of the
resultant two-dimensional velocity vector q , specifically:
and
c c 5/4 4 2 Ve q
hDl/2 (3-53)
The interesting feature of this formulation is that introduction of the
nondimensional dispersion coefficient in Equation 3-51 allows one to
specify the value of the free stream turbulent eddy viscosity. Un-
fortunately, the value of D can vary over two orders of magnitude de-
pending on the geometry of the channel, and in particular, how one
interprets the mechanisms it represents. For example, if one interprets
D as representing a vertical turbulent mixing coefficient, Elder (1959)
shows that it assumes a value of about 0.07. Whereas, if one defines D
to be a lon~itudinal dispersion coefficient for an infinitely wide open
channel its value is approximately 5.9. Between these two extremes,
an entire spectrum of values may be obtained if D is considered to be
a transverse mixing coefficient. Depending on the cross-sectional
34
shape and the longitudinal curvature of the channel investigated, a
compilation of numerous experiments yields values ranging from 0.1 to
1.0 (Fischer 1979). The actual specification of the nondimensional
dispersion coefficient, D , in the present work, will be discussed in
the next section.
Momentum Dispersion Closure
It was suggested in the introduction that existing closure schemes
for the momentum dispersion mechanism lack sufficient theoretical and
experimental justification to warrent direct consideration in the pres-
ent work. However, the general effects of momentum dispersion can be
indirectly investigated in the following way. First, the enhanced
mixing due to vertical shear can be included in the (k-e) turbulence
closure model by appropriate specification of the nondimensional dis-
persion coefficient, D . In order to compare the effects of the spec-
ified momentum dispersion coefficient on the simulation results, a range
of 0.2 < D < 5 will be examined. Secondly, momentum transport due to
curvature of the depth mean streamlines can be incorporated into the
standard (k-e) turbulence model by modifying the value of CV in equa-
tion (3-38) when the streamline curvature is large. The approach used
in the present study is an approximation to a method, based on an alge-
braic Reynold's stress model, presented by Leschziner and Rodi (1981).
The necessity for including the streamline curvature modification, and
its implementation is discussed in Chapter 6.
CHAPTER 4
COMPUTATIONAL METHOD
The numerical technique employed to obtain approximate solutions
for the depth-integrated model equations is the third-order accurate
QUICK (Quadratic Upstream Interpolation for Convective Kinematics)
method developed by Leonard (1979). In this chapter, the derivation
of QUICK which is based on a conservative, control volume integral
formulation will be discussed. In addition, the inherent convective
stability of the technique and the absence of classical numerical
diffusion (Roache 1972) will be illustrated by comparison of QUICK with
central and upwind differencing.
Control-Cell Formulation
To apply QUICK in a depth-integrated transport model, each equa-
tion must be integrated over its appropriate control cell on a constant
space, staggered, square computation grid (Figure 5) and in time. For
purposes of generality, it is first convenient to cast all of the model
equations into the following vector form:
where
aFt + aFx + aFy = G at ax ay
h
Uh Ft = Vh
kh
35
(4-1)
Fx =
Fy =
and
' G =
36
Uh 2
UUh + ~ -. 2 UVh - T xy
A
Ukh a - v -t ax
Ueh Vt a - --ae ax
Vh UVh - T xy
2
T xx
(kh)
(Eh)
VVh + _gg_ - T 2 yy
Vkh -
Veh -
0
azb gh - + ax tbx
azb gh - + ay tby
Ph + Pk - eh
A
:-cclh - c2eh) + p
k e
in which all variables have been defined previously. The exact control
cell integration of equation (4-1) may then be written:
37
( n+l Ft
~t 6x ~
dt = J f 2 r G dy dx dt
0 _!Si -~ 2 2
(4-2)
Noting that steady-state solution are sought, a simple Euler time
integration is adopted which allows equation (4-2) to be rewritten as
follows:
(4-3)
in which quantities in parentheses with subscripts R , L T , and
B denote right, left, top, and bottom cell face averages, respectively,
and underlined quantities denote cell averages (Figure 6).
Derivation of QUICK
The cell and cell face averages that are required in equa-
tion (4-3) are approximated by integrating a six-point upstream weighted
quadratic interpolation function over the appropriate limits. To illus-
trate this procedure, consider the computation the right cell face aver-
age of a field variable, f , using the information provided in figure 7.
In this case, the U and V velocity components are both positive and
directed to the right and up, respectively. Combining a Newton forward
38
difference interpolation formula in the longitudinal, ~ , direction
with a Gauss backward difference interpolation formula in the transverse,
~ , direction, a quadratic interpolation function is constructed which
reads (Hildebrand 1956):
+ .!.c~2 + ~)f + (~ - !l + ~~\f - ~~f 2 0,1 2 2 ; 0,-1 1,-1 (4-4)
where ~ = x/~s , and ~ = y/~s are local nondimensional coordinates.
The right cell face average is then computed as follows:
(f) = R
1 t 1 2
( 4-5) .
Due to the symmetry of the integration operation, a change in
the sign of the V velocity component will not alter equation (4-5).
However, a change in the sign of the U velocity component will re-
quire that the mirror image of figure 7 be used with the indices
shifted to the right by one. Performing the integration prescribed
in equation (4-5) results in the following interpolation formula:
39
(4-6)
The purpose of this adjustment is to maintain the upstream weighting
of the interpolation formula. This all important feature of the tech-
nique is the subject of the next section.
Concept of Convective Sensitivity
By means of a simple analysis in one dimension, it is easily
shown that the upstream weighting of the interpolation formula solely
accounts for the impressive convective stability of QUICK. Following
Leonard (1979), consider the pure convection of a scalar function $
moving positively from left to right through a control cell centered
at the origin (x = 0). Noting figure 8, the influx to the control
cell is given by
INFLUX (4-7)
Defining the convective sensitivity (C.S.) of a finite difference
approximation to be
c.s. (4-8)
it is clear that, for convective stability, the convective sensitivity
must always be negative. In effect this is a boundedness condition
40
which requires the influx to the control cell to decrease as $0 in-
creases toward an asymtotic value, and visa versa. Without transverse
variation, the convective sensitivity of QUICK is written:
c.s. QUICK
which upon simplification becomes:
(4-10)
Thus, it is apparent that QUICK maintains convective stability given
that:
(4-11)
If the upstream weighted curvature term is omitted, thereby reducing the
order of interpolation from quadratic to linear, the familiar central
differencing formula results. The convective sensitivity for central
differencing is written as follows:
c.s. CENTRAL (4-12)
which implies that central differencing maintains convective stability
when
41
(4-13)
For completeness, if a zero order interpolation is adopted (i.e.
upwind differencing) the convective sensitivity becomes
C.S. UR = (4-14) Upwind ~s
which suggests that upwind differencing always maintains convective
stability.
Comparison of equations (4-11), (4-13), and (4-14) clearly illus-
trates the stabilizing effect of upstream weighting of the interpolation
formula. At one extreme, it is seen that central differencing which
does not upstream weight the interpolation has an intolerable convec-
tive stability restriction. Whereas at the other end of the spectrum,
upwind differencing which is entirely upstream weighted is without a
convective stability restriction.
Truncation Error Analysis
With respect to convective stability, upwind differencing has been
shown to be far superior to both QUICK and central differencing. How-
ever, the general applicability of upwind differencing in practical
computation is often obviated by the effects of first order truncation
errors (i.e. numerical diffusion). In order to examine the formal
truncation error of the QUICK approximation for the convection term,
consider again the problem of pure convection but with the velocity
assumed uniform and equal to one. The convective flux is then written
42
(4-15)
Substituting the appropriate QUICK formulas for ~ and $1 yields:
(4-16)
The truncation error for this approximation could be obtained by
examining the remainder term after taking the difference of the two
interpolation functions. However, an alternative approach is to ob-
tain the truncation error by simply deriving the finite difference
approximation by means of Taylor series exspansions. Writing "' "'-2 <!>_ 1 , and $1 as a Taylor series about $0 one obtains:
2 4 3 II I = "' - Us"'' + 2b.s "'" - -As <I> 't'Q 't'Q "'O 3 0
2 4 I II I + 3b.s <1>0 + h.o.t. (4-17)
2 3 "' ="' - 6s<j>' +As <I>'' - as <I>' I I "'-1 "'O 0 2 0 6 0
4 + 8S n.t I I I + h t 24'1'0 .o .. (4-18)
2 3 "' = "' + A "'' + bas "''' + As "'''' '1'1 "'O s't'Q 2 "'O 6 't'Q
4 + ~II II + h t 24'1'0 .o .. (4-19)
43
Combining the appropriate scalar multiples of equations (4-17)-(4-19)
and dropping all higher order terms results in:
(4-20)
which upon rearrangement becomes:
(4-21)
The formal third order accuracy exhibited in equation (4-21) shows that
QUICK is entirely free of what is classically called numerical diffusion.
Simplifying Assumption
Strict application of the two-dimensional QUICK method in obtain-
ing approximations for the required cell and cell face averages in
equation (4-1) is not a simple task. The appearance of products of
field variable would necessitate the evaluation of integrals of products
of interpolation function. With the occurrence of triple products in
the convection terms, it is obvious that the amount of computational
effort would be prohibitive. Consequently, it is necessary to assume
that averages of products of field variables are reasonably well approx-
imated by the products of the individual field variable average. For
example, computation of the right cell face average for the square of a
field variable would proceed as follows. First the field variable is
decomposed into its cell face average (f)R and a variation along the ~
cell face f~ where
and thus
1
f 2 1 2
44
= (f)~ +
1
f 2 1 2
(4-22)
Then, as a matter of convenience, the integral on the right-hand side
is assumed small and neglected.
CHAPTER 5
MODEL SIMULATIONS
The test problems chosen for model simulation were steady, free
surface flow in a wide, shallow, rectangular channel with and without
an abrupt expansion in the width. A systematic program of numerical
experimentation was performed in two phases. In the first phase, at-
tention was focused on evaluation of the QUICK finite difference scheme
as applied to the solution of the frictionally damped shallow water wave
equations. Consequently, turbulence closure was neglected so that the
convective stability of QUICK could be examined without the aid of the
stabilizing effects of turbulent diffusion/dispersion. The second phase
of computation consisted of essentially repeating the two test problems
with the (k-e) turbulence model incorporated in the hydrodynamic code.
In this chapter, the details of the numerical experiments are presented.
Simulations Without Turbulence Closure
The model parameters chosen for the first phase of computation
were a longitudinal channel slope of 0.005, a transverse channel slope
of zero, a friction coefficient of 0.018, and an upstream normal depth
of 4.73 ft which corresponds to an arbitrarily chosen flow rate per unit
width of 30 ft3/sec/ft.
Backwater Computation
For the channel simulation without an abrupt expansion, the flow
45
46
was required not to vary in the transverse direction which allowed a
comparison of the computed water surface profile with a standard fourth-
order Runge Kutta solution (Hildebrand 1956 and White 1974) of the well
known gradually varied flow equation (Chow 1959).
The water surface elevation at the downstream boundary was arbi-
trarily fixed at a depth of 4.0 ft and the outflow velocity was computed
using the mass conservation equation. Uniform flow was specified at the
upstream boundary by requiring that the longitudinal gradients of the
water surface elevation and the velocity equal zero. Using 100 grid
points in the instream direction, of which half represented depth and
half represented velocity locations, it was found that a grid spacing
of 90.0 ft yielded a satisfactory approximation for uniform flow at the
upstream boundary.
At the outset of the computation, all interior depth points were
set equal to the upstream normal depth, and all velocity points were set
to zero.
Channel Expansion Simulation
The abrupt channel expansion simulation was performed with an
outlet to inlet channel width ratio of 1.5. In this computation, both
velocity components and the normal gradient of the water surface eleva-
tion were specified to be zero at rigid walls. This was accomplished
by defining image points outside of the computational domain such that
interpolated values of cell averages or cell face averages of the
velocity components, and the normal gradients of the water surface
elevation that coincided with the wall boundary were exactly zero.
47
Subsequently, the position of the channel wall was chosen such that
grid points for the transverse velocity component resided on the wall
boundary (Figure 9), and grid points for the streamwise velocity compo-
nent and water surface elevation were located one half of a grid spacing
from the channel wall (Figure 10).
Symmetry was imposed at the channel center line by employing a
reflection boundary condition. Here the normal gradients of the
streamwise velocity component and the water surface elevation were set
equal to zero. The treatment of the transverse velocity component is
identical to that for a rigid wall boundary.
The upstream and downstream boundaries were positioned suffi-
ciently far from the abrupt expansion so that the requirement of uniform
flow (i.e., zero gradient) was met when a grid spacing of 90.0 ft was
employed.
An essentially arbitrary initial condition was again used where
all of the depths were set equal to the upstream normal depth, and all
velocities were set to zero.
Simulations With Turbulence Closure
Wall Boundary Layer Computation
The first test problem performed using (k-e) turbulence closure
was the simulation of a fully developed turbulent wall boundary layer.
This test simulation was chosen to examine the performance of the wall
boundary formulation by comparison with available experimental data
(Nakagawa et al. 1975).
48
The experimental setup of Nakagawa et al. was approximated by
adopting a channel half width of 0.25 ft with the free surface and
channel bottom assumed to be parallel and stress free. Consequently,
the longitudinal gradients of the nontrivial field variables U, k, and
e, and the friction coefficient were set to zero. In other words, the
test problem was effectively reduced to a one-dimensional simulation of
flow between two parallel plates with boundary condition required only
at the channel center line and wall.
A reflection boundary was employed at the channel center line by
setting the normal gradient of U, k, and e to zero.. For convenience,
the center-line boundary was located at the midpoint between the first
two grid points.
The wall boundary was approximated by the "wall function" method
of Launder and Spalding (1972). The beauty of wall function method is
that the shear stress is matched at the wall boundary with the interior
flow field, which obviates the need to resolve the near wall boundary
layer. Thus, a much coarser grid may be used near the wall.
To apply the wall function method, it is necessary to assume that
the near wall region is in local equilibrium so that the shear stress
is constant and the logarithmic velocity law is valid. Thus, the shear
velocity, U*' may be related to the local velocity, Uw' defined at the
nearest interior grid point, by
KU w (5-1)
49
where again, the location of the wall boundary is defined between grid
points. Recall that, by definition, the wall shear stress per unit mass
is written:
(S-2)
which suggests that the wall shear stress could, in fact, be obtained
from equation (S-1). However, an interesting feature of the wall func-
tion method is the desire to build as much interior field information
as possible into the wall shear stress equation. Consequently, equa-
tion (3-34) is used to recast equation (S-2) into the form:
t w = cl/4 kl/2 u
v w * which upon substitution of equation (S-1) yields:
(5-3)
(S-4)
A favorable attribute of equation (S-4) is that it preserves the sign of
the wall shear stress vector.
Another interesting feature of the wall function method is that
the near wall energy dissipation rate per unit mass, e , is specified w as a function of the local turbulence kinetic energy, k . The relation-w ship used to fix e is obtained by sµbstituting equation (3-34) into a w local form of equation (3-30), namely,
50
c3/4 k3/2 v w
£ = w K(l:ly/2) (5-5)
The near wall values of the velocity, U , and the turbulence kinetic w
energy per unit mass, k , are computed using their respective balance w
equations, but with the following modifications: (1) the normal
gradient of U and k at the wall are set to zero, (2) the usual shear
stress term is replaced by the computed wall shear stress, i.e.,
(5-6)
and (3) the dissipation term in the turbulence energy equation is modi-
fied to account for the large increase in dissipation rate in the near
wall region, specifically:
£ = 1 l:ly l:ly - zo J
z 0
£ dy w
which with some simplification becomes:
£ = c314k3/ 2 ln (f:ly/z ) v w 0
Kf:ly
(5-7)
(5-8)
The initial condition used in the wall boundary layer computation
corresponded to a simple plug flow. Specifically, the initial velocity
distribution across the channel was assumed uniform and equal to 0.5 ft/
sec, with arbitrary, but nonzero, values specified for the turbulence
quantities k and £ .
51
Channel Expansion Simulations
The final set of test simulations with (k-e) turbulence closure
were performed using a wide, rectangular open channel with an abrupt
expansion in width. An area ratio of 1.45 was adopted for purposes of
(1) insuring the validity of a centerline symmetry boundary, and
(2) minimizing the number of superfluous grid points in the free stream.
The physical parameters specified for this series of computations
were a longitudinal channel slope of 0.0005, a transverse channel slope
of zero, and a friction coefficient of 0.0045 which corresponds to a
roughness height, z , of 0.005. 0
This combination of parameters results
in a flow that is both fully turbulent and rough, which justifies the
use of the quadratic shear stress law, and the logarithmic velocity law.
The nondimensional dispersion coefficient was specified at a
nominal value of one for the initial set of test simulations, and sub-
sequently, changed to 0.2 and 5 for purposes of examining its effect on
the model results.
Wall boundaries were again approximated using the wall function
technique; with a minor modification required to handle the convex
corner of the abrupt exspansion. In the corner region, the top cell
face of a U velocity cell, and the left cell face of a V velocity
cell are divided equally between a wall and a fluid segment (Figure 11).
Consequently, only one half of the predicted wall shear stress is used
in the computation of these points.
The water surface elevation at the upstream boundary is specified
to have zero gradient in the transverse direction, zero curvature in
52
the longitudinal direction, and a constant depth of 5.0 ft. This con-
figuration corresponds to a unidirectional inflow boundary in which the
transverse velocity component, V , is set to zero, and the longitudinal
velocity component, U , is obtained from continuity. In addition, the
turbulence quantities kh and Eh at the upstream boundary are assumed
to have zero curvature in the longitudinal direction.
The downstream boundary configuration is essentially identical to
the upstream boundary with the exception that the depth was set to 4.9 ft
for purposes of accelerating convergence.
An initial condition for the (k-E) turbulence closure simulations
was generated by running a constant eddy viscosity version of the model
to an approximate steady state. A steady state, constant eddy viscosity
of 2.0 ft2/sec was found to be consistent with the earlier assumption on
the value of the nondimensional dispersion coefficient (i.e., D = 1.0).
An arbitiary initial condition was used in the constant eddy viscosity
computation with the interior water surface elevations set to 5.0 ft,
and the velocity components set to zero.
CHAPTER 6
RESULTS AND DISCUSSION
In this chapter, the results of the model simulations described in
Chapter 5 will be presented and discussed.
Results of Simulations Without Turbulence Closure
Backwater Computation
The simple straight wall channel simulation was performed using a
stable time step of .75 sec. which resulted in an average one-dimensional
Courant number [= (q + /iJ.i.) at/as] of about .15. An arbitrarily chosen
maximum difference, between the QUICK and Runge-Kutta solutions, of
.01 percent was achieved in 540 iterations. Consequently, the comparison
of the predicted water surface profiles provided in figure 12 shows ex-
tremely good agreement.
Expansion Channel Simulation
The second test simulation, which simply included an abrupt expan-
sion in width, required 3500 iterations with a time step of .4 sec. to
yield a steady solution. The convergence criteria used in this simula-
tion was to require that the computed flowrates at the upstream and
downstream boundaries differed by less than .01 percent. An average two-
dimensional Courant number [= (q + ~2gh) at/as] was found to be about
.1, and the CPU time required was approximately 12 min on an IBM 3032.
It is interesting to note that much of the required CPU time was
53
54
consumed "washing out" the rather arbitrary initial condition specified.
As a result, a companion simulation was made using an initial water sur-
face profile that decreased linearly in the vicinity of the expansion.
In this run, the convergence criteria was satisfied in about 6 min. of
CPU time.
The results of the abrupt expansion calculation are presented in
figure 13, a vector plot of the depth-integrated resultant velocities, and
figure 14, a three dimensional plot of the water surface elevations.
The significance of the results presented becomes readily apparent
when one attempts to repeat the test computations using linear interpo-
lation for the cell and cell face averages, i.e. central differencing.
In trial calculations of the two test problems, central differencing
failed to produce steady solutions in either case. The inability of
the central difference procedure to achieve stable solutions results
from the poor convective stability of the technique which leads to un-
damped error growth (Wiggles).
Stable solutions could have been obtained with the central dif-
ferencing if sufficient eddy diffusion was introduced to satisfy the
cell Reynolds number restriction (Roache 1972):
(6-1)
However, the explicit introduction of "physical" diffusion in the
central difference calculation, for purposes of satisfying the cell
Reynolds number restriction, is essentially equivalent to using upwind
55
differencing, which implicitly introduces numerical diffusion with the
steady state numerical diffusion coefficient given by:
0 =~ num 2 (6-2)
The remarkable similarity seen in equations (6-1) and (6-2) is not a
mere coincidence.
Convective stability gained at the expense of introducing explicit
or implicit numerical diffusion is of little use if ones desire is to
investigate turbulence closure models. Consequently, the use of QUICK
in the present test application of the depth-integrated ·(k-g) turbulence
closure model is clearly supported.
Results of Simulations With Turbulence Closure
Wall Boundary Layer Simulation
The model predictions for the simple wall boundary layer simula-
tion were obtained by repeating the computation with different pressure
gradients (i.e. channel slopes) until a reasonable agreement between the
predicted and experimental (Nakagawa et al. 1975) values of the free
stream velocity, Umax and shear velocity, U* was achieved (Table 1). A
comparison of the model predictions with the experimental data of
Nakagawa et al. (1975) are shown in figure 15, a nondimensional plot of
the velocity defect profile, figure 16, the nondimensional distribution
of the turbulence kinetic energy per unit mass, and figure 17, the
56
Table 1 Comparison of the Experimental and Predicted Values of the Free
Stream Velocity, U , and the Shear Velocity, U* max
Experiment Model (Nakagawa et al. 1975) Prediction
Umax 5.52 x 10-1 6.19 x 10-1 (ft/sec)
u .... 2.66 x 10-2 2.74 x 10-2 (ft/sec)
57
nondimensional distubution of the energy dissipation rate per unit mass.
The close agreement between the model predictions and the experi-
mental data is not surprising when one recalls that the empirical coef-
ficients specified in the (k-e) turbulence model were in fact originally
"tuned" for constant pressure gradient wall boundary layers (Launder and
Spalding 1974).
A final point worth discussion is the discrepency between the pre-
dicted turbulence energy per unit mass and the experimental data (Fig-
ure 16) near the channel centerline (y/h = 1.0). The apparent error re-
sults from the fact that the experimental measurements of Nakagawa et al.
were actually of a vertical boundary layer in an open channel flow, and
subsequently, reflect the presence of a free surface. Rodi (1981)
argues that use of a symmetry boundary condition at the free surface
does not account for the expected reduction in the length scale of tur-
bulence. Consequently, the turbulence energy should in fact be over
predicted in that region. Realizing that the free surface boundary
problem is obviated in a depth integrated simulation model, no attempt
was made at incorporating the recommended boundary correction (Rodi
1981) into the wall boundary computation.
Channel Expansion Simulation
The final set of simulations included (k-e) turbulence closure in
the channel expansion problem. In order to generate a region of recir-
culation in the constant eddy viscosity starting solution, it was neces-
sary to adopt a grid spacing of 5.0 ft so that the value of the convec-
tive acceleration and turbulent transport terms were of the same order
58
of magnitude as the bottom friction terms. When grid spacings much
larger than 5.0 ft were tested, recirculation was almost entirely
inhibited.
Using a grid spacing of 5.0 ft in the simulation with (k-e) turbu-
lence closure, 4000 iterations with a stable time step of .01 sec were
required to achieve a steady solution. The convergence criteria used in
these simulations were that (1) the computed flowrates at the upstream
and downstream boundaries differed by less than .1 percent, and (2) the
distubutions of the turbulence quanties kh and eh, which were initialized
at arbitrarily small, but nonzero, values were time invariant. With an
average Courant number of about .OS, each simulation required approxi-
mately 140 min of CPU time.
The results of the channel expansion calculation with (k-e) clo-
sure are presented in figure 18, a contour sketch of the water surface
elevations, figure 19, a vector plot of the depth-averaged resultant
velocities, and figures 20, 21, and 22, a comparison of predicted
longitudinal velocity profiles at three locations with the experimental
measurements of Moss et al. (1977).
The aspect ratio (h/W ) of the inlet channel was approximately .09 0
which suggests that one should expect a nondimensional reattachment
length (xr/W1) in the range of 4.5 to 5.0 (Figure 3). However, it is
seen in figure 19 that the predicted reattachment length is about 3.2,
which corresponds to an error of 30 percent.
In order to compare the predicted velocity profiles with experi-
mental measurements, it was first necessary to 1) rescale the
59
nondimensional x-axis such that the origin corresponded to the reattach-
ment point, xr/W1, thereby removing the problem of variable reattachment
lengths, and 2) normalize the predicted velocity distribution so that
predicted and measured velocities in the free stream matched.
The agreement between the model predictions and experimental
measurements within the region of recirculation (Figure 20) is satis-
factory, however, the velocity profiles in the region bounding the reat-
tachment point are noticably underpredicted (Figures 21 and 22).
The overall poor agreement between model predictions and experi-
mental measurements is directly attributable to the overprediction of
the depth-averaged eddy viscosity in the region of strong streamline
curvature (Leschziner and Rodi 1981). In an attempt to improve the
model predictions, an approximation to the streamline curvature modifi-
cation of Leschziner and Rodi (1981) was employed. In their work,
Leschziner and Rodi showed that the value of the coefficient CV in
equation (3-38) is significantly less than .09 in regions of strong
streamline curvature. Consequently, as a first approximation, the value
of CV was decreased to .03 at all grid points interior to the wall func-
tion grid points in the region downstream of the step for a distance of
6W1. The value .03 was chosen on the basis that it represents a reason-
able average over the distribution of C computed by Leschziner and Rodi v (1981).
A comparison of the nondimensional depth-averaged eddy viscosity
distributions computed with and without the ad hoc curvature modifica-
tion is shown in figure 23. From a symptomatic point of view, the
60
curvature modification does, in fact, overcome the problem of overpredic-
tion of the depth averaged eddy viscosities, however, the degree of
success can only be measured by the improvement upon the predictions for
the reattachment length and the velocity profiles.
Examination of figure 24, a vector plot of the depth averaged
velocities, reveals a significant increase in the nondimensional reat-
tachment length xr/W1, resulting from curvature modification. The pre-
dicted value in this simulation is approximately xr/W1 = 4.6 which
agrees well with the experimental measurement depicted in figure 3. In
addition, a marked improvement is seen in the comparison of the velocity
profiles in the region near reattachment (Figures 27 and 28).
The final set of test simulations were performed to investigate
the effects of varying the nondimensional dispersion coefficient, D, on
the model predictions. In addition to the previously described simula-
tions with D equal to one, two more test computation, both of which
included curvature correction, were run with D equal to 0.2 and 5.0,
respectively.
For the case where the nondimensional dispersion coefficient was
reduced to 0.2, the model responded with a 5 percent increase in the re-
attachment length, and a noticable, but slight increase in the velocity
gradient across the recirculation zone (Figure 29). On the other hand,
when D was increased to 5.0 the reattachment length decreased by 5 per-
cent with a corresponding decrease in the velocity gradient across the
recirculation region.
Insight into the proper specification of the nondimensional
61
dispersion coefficient, D, can be gained by examining figure 30, a com-
parison of the computed distributions of the depth-integrated eddy
viscosity at the upstream boundary for varying values of D.
It is clear that when D is set to 0.2, the magnitude of the free
stream eddy viscosity is underpredicted. This observation is easily
argued on the basis that the near wall value of the eddy viscosity,
which is fixed by the wall function, can not physically take on a value
twice that of the free stream. Thus, a nondimensional dispersion coef-
ficient of 0.2 appears to be too small.
Experimental measurements of flows in wide, rectangular channels
(Rodi 1980) suggest that the wall boundary layer thickness is less than
20 percent of w1, the channel half width. As a result, it seems reason-
able that the depth-averaged eddy viscosity over 80 percent of the
channel should be constant and solely dependent on bottom effects.
Although this feature is represented well when D is equal to 0.2 and
1.0, respectively, the same can not be said when D is equal to 5.0. In
this case, the depth averaged eddy viscosity varies significantly over
50 percent of the flow, which would suggest that the free stream value
is overpredicted.
Consequently, the nominal value of one, originally chosen for
the nondimensional dispersion coefficient, appears to be the most
satisfactory.
CHAPTER 7
SUMMARY AND CONCLUSIONS
In summary, the objective of the present study has been to develop
a two-dimensional depth-integrated free surface hydrodynamic model
capable of accurately simulating local regions of recirculating flow
induced by abrupt changes in channel geometry. This objective was suc-
cessfully accomplished by attacking the problem in two phases.
The first phase focused on the development and testing of the
depth-integrated hydrodynamic model without the effective stresses con-
sidered. The purpose of this phase was to examine the convective sta-
bility of QUICK under conditions where the smoothing effects of physical
diffusion/dispersion were unimportant. Based on the numerical results
presented, it is concluded that:
1. When applied to highly convective flow problems, QUICK is far
superior to either central or upwind differencing in that it offers a
high degree convective stability, but is free from classical numerical
diffusion.
2. QUICK is ideally suited for use in basic research directed
toward examining effective stress closure schemes.
The second phase of research concentrated on building adequate
closure for the effective stresses into the hydrodynamic model. In
this phase, attention was primarily focused ori closure of the depth-
integrated Reynolds stresses by means of the (k-e) turbulence closure
scheme. Although there was no direct attempt made at momentum dispersion
62
63
closure, the effects of momentum dispersion were examined indirectly, by
investigating of the effects of streamline curvature, and the magnitude
of the nondimensional dispersion coefficient on turbulence closure.
Based on the test application of the (k-e) turbulence closure model to
the problem of separation in a wide, shallow rectangular channel with an
aburpt expansion in width, the following conclusions are presented:
1. Use of the standard depth-integrated (k-e) turbulence model
resulted in an underprediction of 30 percent in the reattachment length,
and a poor representation of velocity profile measurements near the
reattachment point.
2. When an ad hoc curvature correction was applied in the (k-e)
scheme, a marked improvement in the model predictions resulted. Con-
sequently, the effects of curvature of the depth mean streamlines has
been shown significant, wrrich suggests that future work is needed to
differentiate between the contribution of momentum dispersion due to
streamline curvature, and the need for curvature correction in the (k-e)
turbulence model.
3. Variation of the nondimensional dispersion coefficient, D,
over a physically reasonable range of values resulted in noticeable
changes in the predicted reattachment length and the depth-averaged
velocity profiles. However, based on results presented herein, it
must be concluded that a value of one is the most appropriate of those
examined.
Model Limitations and Recommendations for Future Research
The fundumental problem which limits the general utility of the
64
present version of the model is it~ basic spatial and temporal structure.
The spatial structure of the code was designed using a constant
space, square grid cell, which requires a uniform distribution of grid
points over the computational domain. Consequently, flow simulations
such as the channel expansion problem require an unusually large number
of grid points to insure that the position of upstream, downstream, and
free stream boundaries does not affect the solution in the region of
interest.
In order to overcome this problem, the spatial structure of the
code could be modified to (1) accept a simple variable grid spacing, or
(2) transform the original governing equations by either an exponential
stretch function or boundary-fitted coordinates. Any of the foremen-
tioned alternatives would offer one the latitude to adjust the grid
density where necessary, however, care must be taken to insure that the
resulting quadratic interpolation functions are constructed and used
properly.
The problem associated with the time step structure is that an
explicit update procedure has been employed, which is bound by very
restrictive temporal stability criteria. Consequently, the model is
rather slow to converge to a steady state solution. Alternative methods
for rectifying this problem are (1) retain the transient sense of the
solution procedure and develop a semi-implicit or fully implicit tempo-
ral update, or (2) drop the temporal terms and solve the resulting set
of nonlinear matrix equations by the Newton-Raphson method.
Finally, it should be pointed out that the QUICK technique was
65
originally developed to solve steady state problems. If ones desire
is to address transient phenomena then quadratic upstream interpolation
must also be applied to the temporal terms, which results in the QUICKEST
(Quadratic Upstream Interpolation for Convective Kinematics with Esti-
mated Streaming Terms) method, which is third-order in both space and
time. A complete description of QUICKEST is presented by its originator,
Leonard (1979).
REFERENCES
Abbott, D. E., and S. J. Kline, "Theoretical and Experimental Investiga-tion of Flow Over Single and Double Backward Facing Steps," Report MD-5, Thermosciences Division, Department of Mechanical Engineering, Stanford University, June 1961.
Abbott, D. E., and S. J. Kline, "Experimental Investigation of Subsonic Turbulent Flow Over Single and Double Backward Facing Steps," Journal of Basic Engineering, Transactions, ASME, Vol. 84, 1962.
Abbott, M. B., "A Pathology of Mathematical Modeling," Journal of Hydrau-lic Research, Vol. 14, No. 4, 1976.
Abbott, M. B., and C.H. Rasmussen, "On the Numerical Modeling of Rapid Expansions and Contractions in Models that are Two-Dimensional in Plan," Paper A104, 17th !AHR Congress, Baden-Baden, Germany, 1977.
Abbott, M. B., Computational Hydraulics: Elements of the Theory of Free Surface Flows, Pitman Publishing Limited, London, 1980.
Atkins, D. J., S. J. Maskell, and M.A. Patrick, "Numerical Predictions of Separated Flows," Numerical Methods in Engineering, Vol. 15, No. 1, 1980.
Bradshaw, P., and F. Y. F. Wong, "The Reattachment and Relaxation of a Turbulent Shear Layer," Journal of Fluid Mechanics, Vol. 52, No. 1, 1972.
Bradshaw, P., "Review of 'The Calculation of Turbulent Boundary Layers on Spinning and Curved Surfaces,' by B. E. Launder et al.," Journal of Fluids Engineering, Transactions, ASME, Vol. 99, No. 2, 1977.
Castro, I. P., "Numerical Difficulties in the Calculation of Complex Tur-bulent Flows," Proceedings, Symposium on Turbulent Shear Flow, Pennsyl-vania State University, Vol. 1, 1977.
Chapman, R. S. and C. Y. Kuo, "Application of a High Accuracy Finite Difference Technique to Steady Free Surface Flow Problems," Presented at the Joint ASME/ASCE Mechanics Conference, Boulder CO, June 1981.
Chow, V. T., Open Channel Hydraulics, McGraw-Hill Book Co., Inc., New York, 1959.
Davis, T. W., and D. J. Snell, "Turbulent Flow Over a Two-Dimensional Step and Its Dependence Upon Upstream Flow Conditions," Proceedings, Symposium on Turbulent Shear Flows, Pennsylvania State University, Vol. 1, 1977.
66
67
Durst, F., and A. K. Rastogi, "Theoretical and Experimental Investiga-tions of Turbulent Flow With Separation," Proceedings, Symposium on Tur-bulent Shear Flows, Pennsylvania State University, Vol. 1, 1977.
Durst, F., and C. Tropea, "Turbulent, Backward-Facing Step Flows In Two-Dimensional Ducts and Channels," Proceedings, Third Symposium on Turbulent Shear Flows, University of California, Davis, Sept. 1981.
Eaton, J. K., and J.P. Johnston, "A Review of Research on Subsonic Turbulent Flow Reattachment," AIAA Journal, Vol. 19, No. 9, 1981.
Elder, J. W., "The Dispersion of Marked Fluid in Turbulent Shear Flow," Journal of Fluid Mechanics, Vol. 5, No. 4, 1959.
Engelund, F., Flow and Bed Topography in Channel Bends, Journal of the Hydraulics Division, ASCE, Vol. 100, No. HYll, 1974.
Etheridge, D. W., and P.H. Kemp, "Measurements of Turbulent Flow Down-stream of a Rearward~Facing Step," Journal of Fluid Mechanics, Vol. 86, No. 3, 1978.
Fischer, H.B., E. J. List, R. C. Y. Koh, J. Imberger, and N. H. Brooks, Mixing in Inland and Coastal Waters, Academic Press, Inc., New York, 1979.
Flokstra, C., "Generation of Two-Dimensional Horizontal Secondary Currents," Delft Hydraulics Laboratory, Report S 163-II, 1976.
Flokstra, C., "The Closure Problem for Depth Averaged Two-Dimensional Flows," Paper A106, 17th LAHR Congress, Baden-Baden, Germany, 1977.
Garde, R. J., K. G. Ranga Raju, and R. C. Mirshra, "Subcritical Flow in Open Channel Exspansions," Irrigation and Power, Vol. 36, No. 1, 1979.
Gosman, A. D., W. M. Pun, A. K. Runchal, D. B. Spalding, and M. Wolfshtein, Heat and Mass Transfer in Recirculating Flows, Academic Press, London, 1969.
Han, T., J. A. C. Humphrey, and B. E. Launder, "A Comparison of Hybrid and Quadratic-Upstream Differencing in High Reynolds Number Elliptic Flows," Computer Methods in Applied Mechanics and Engineering, Vol. 29, 1981.
Hanjalic, K., and B. E. Launder, "Turbulence in Thin Shear Flows," Journal of Fluid Mechanics, Vol. 52, No. 4, 1972.
Harlow, F. H., and P. I. Nakayama, "Transport of Turbulence Energy Decay Rate," Los Alamos Scientific Laboratory Report LA-3854, University of California, 1968.
68
Harlow, F. H., and C. W. Hirt, "Generalized Transport Theory of Aniso-tropic Turbulence," Los Alamos Scientific Laboratory Report LA 4086, University of California, 1969.
Hildebrand, F. B., Introduction to Numerical Analysis, McGraw-Hill Book Co., Inc., New York, 1956.
Hinstrup, P., A. Kej, and U. Kroszynski, "A High Accuracy Two-Dimensional Transport-Dispersion Model for Environmental Applications," Paper B17, 17th !AHR Congress, Baden-Baden, Germany, 1977.
Hinze, J. 0., Turbulence, McGraw-Hill Book Co., Inc., New York, 1959.
Hirsh, R. S., "Higher Order Accurate Difference Solutions of Fluid Mechanics Problems by Compact Difference Techniques," Journal of Compu-tational Physics, Vol. 19, No. 1, 1975.
Irwin, H. P. A. H., and P. Arnot Smith, "Prediction of the Effects of Streamline Curvature on Turbulence," The Physics of Fluids, Vol. 18, No. 6, 1975.
Holly, F. M., and A. Preissman, "Accurate Calculations of Transport in Two Dimensions," Journal of the Hydraulic Division, ASCE, Vol. 103, No. HYll, 1977.
Kim, J., S. J. Kline, and J. P. Johnston, "Investigation of Separation and Reattachment of a Turbulent Shear Layer: Flow Over a Backward-Facing Step," Report MD-37, Thermosciences Division, Department of Mechanical Engineering, Stanford University, April 1978.
Kuipers, J., and C. B. Vreugdenhil, "Calculation of Two-Dimensional Hori-zontal Flow," Report S 163-1, Delft Hydraulics Laboratory, 1973.
Laufer, J., "Investigation of Turbulent Flow in a Two-Dimensional Channel," Report 1053, NACA, 1951.
Launder, B. E., and D. B. Spalding, Lectures in Mathematical Models of Turbulence, Academic Press, Inc., New York, 1972.
Launder, B. E., and D. B. Spalding, "The Numerical Calculation of Turbu-lent Flows," Computer Methods in Applied Mechanics and Engineering, Vol. 3, 1974.
Launder, B. E., C. H. Pridden, and B. I. Sharma, "The Calculation of Turbulent Boundary Layers on Spinning and Curved Surfaces," Journal of Fluids Engineering, Transactions, ASME, Vol. 99, No. 1, 1977.
69
Lau, Y. L., and B. G. Krishnappan, "Ice Cover Effects on Stream Flows and Mixing," Journal of the Hydraulics Division, ASCE, Vol. 107, No. HYlO, 1981.
Lean, G. H., and T. J. Weare, "Modeling Two-Dimensional Circulating Flows," Journal of the Hydraulic Division, ASCE, Vol. 105, No. HYl, 1979.
Leenderste, J. J., "Aspects of a Computational Model for Long-Period Water Wave Propagation," RM-5297-DR, The Rand Corporation, Santa Monica, CA, May 1967.
Leonard, A., "Energy Cascade in Large-Eddy Simulations of Turbulent Fluid Flow," Advances in Geophysics, 18A, 1974.
Leonard, B. P., M.A. Leschziner, and J. McGuirk, "Third-Order Fin1te-Difference Method for Steady Two-Dimensional Convection," Proceedings of the International Conference on Numerical Methods in Laminar and Turbu-lent Flow, Swansea, July 1978.
Leonard, B. P., "A Stable and Accurate Convective Modelling Procedure Based on Quadratic Upstream Interpolation," Computer Methods in Applied Mechanics and Engineering, Vol. 19, 1979.
Leschziner, M.A., and W. Rodi, "Calculation of Strongly Curved Open Chan-nel Flow," Journal of the Hydraulic Division, ASCE, Vol. 105, No. HYlO, 1979.
Leschziner, M.A., "Practical Evaluation of Three Finite Difference Schemes For the Computation of Steady-State Recirculating Flows," Computer Methods in Applied Mechanics and Engineering, Vol. 23, 1980.
Leschziner, M.A., and W. Rodi, "Calculation of Annular and Twin Parallel Jets Using Various Discretization Schemes and Turbulence-Model Varia-tions," Journal of Fluids Engineering, Transactions, ASME, Vol. 103, 1981.
McGuirk, J. J., and W. Rodi, "A Depth-Averaged Mathematical Model for the Near Field of Side Discharges into Open Channel Flow," Journal of· Fluid Mechanics, Vol. 86, No. 4, 1978.
Mehta, P.R., "Flow Characteristics in Two-Dimensional Expansion," Journal of the Hydraulic Division, ·ASCE, Vol. 105, No. HYS, 1979.
Mellor, G., "Review of 'The Calculation of Turbulent Boundary Layers on Spinning and Curved Surfaces,' by B. E. Launder et al.," Journal of Fluids Engineering, Transactions, ASME, Vol. 99, No. 2, 1977.
70
Militzer, J., W. B. Nicoll, and J. A. Alpay, "Some Observations on the Numerical Calculation of the Recirculation Region of Twin Parallel Sym-metric Jet Flow," Proceedings, Symposium on Turbulent Shear Flows, Penn-sylvania State University, Vol. 1, 1977.
Moss, W. D., Baker, S., and L. J. S. Bradbury, "Measurements of Mean Velocity and Reynolds Stresses in Some Regions of Recirculating Flow," Proceedings, Symposium on Turbulent Shear Flows, Pennsylvania State University, Vol. 1, 1977.
Nakagawa, H., I. Nezu, and H. Ueda, "Turbulence of Open Channel Flow Over Smooth and Rough Beds," Proceedings, Japan Society of Civil Engineers, No. 241, 1975.
Patankar, S. V., and D. B. Spalding, Heat and Mass Transfer in Boundary Layers," Intertext Publishers, London, England, 1970.
Ponce, V. M., and S. B. Yabusaki, "Modeling Circulation in Depth-Averaged Flow," Journal of the Hydraulics Division, ASCE, Vol. 107, No. HYll, 1981.
Rastogi, A., and W. Rodi, "Prediction of Heat and Mass Transfer in Open Channels," Journal of the Hydraulic Division, ASCE, Vol. HY3, 1978.
Reynolds, W. C., "Computation of Turbulent Flows," Annual Review of Fluid Mechanics, Vol. 8, 1976.
Roache, P. J., Computational Fluid Dynamics, Hermosa Publishers, Albu-querque, NM, 1972.
Rodi, W., Turbulence Models and Their Application in Hydraulics: A State of the Art Review, Presented by the !AHR-Section on Fundamentals of Division II Experimental and Mathematical Fluid Dynamics, Delft, 1980.
Rozovskii, I. L., Flow of Water in Bends of Open Channels, Academy of the Sciences of the Ukranian SSR, Kieu, USSR (translation no. OTS 60-51133), Office of Technical Services, Washington, D.C., 1957.
Schamber, D.R., and B. E. Larock, "Numerical Analysis of Flow in Sedi-mentation Basins," Journal of the Hydraulics Division, ASCE, Vol. 107, No. HYS, 1981.
Schlichting, H., Boundary Layer Theory, McGraw-Hill Book Co., Inc., New York, 1955.
Smyth, R., "Turbulent Flow Over a Plane Symmetric Sudden Expansion," Journal of Fluids Engineering, Transactions, ASME, Vol. 101, 1979.
Tennekes, H., and J. L. Lumley, A First Course in Turbulence, MIT Press, Cambridge, Massachusetts, 1972.
71
Townsend, A. A., The Structure of Turbulent Shear Flow, Cambridge Univer-sity Press, London, 1956.
van Bendegom, L., "Some Considerations on River Morphology and River Improvement," De Ingenieur, Vol. 59, B 1-11, 1947.
White, F. M., Viscous Fluid Flow, McGraw-Hill Book Co., Inc., New York, 1974.
NOMENCLATURE
Alphabetic Symbols
c = nondimensional bottom friction coefficients
C = Courant number
c1 = constant equation (3-24)
c2 = constant equation (3-24)
CV = constant equation (3-16)
D = nondimensional dispersion coefficient
ft,~ = quadratic interpolation function
Ft = vector of quantities operated on by the partial derivative with re-spect to time
Fx = vector of quantities operated on by the partial derivative with re-spect to the longitudinal coordinate direction, x
Fy = vector of quantities operated on by the partial derivative with re-spect to the transverse coordinate direction, y
g. = acceleration due to gravity (0, O, - g) 1
G = vector of source quantities
h = water depth
k = turbulence kinetic energy per unit mass
1 = turbulent macroscale
p = pressure
P' = turbulent pressure fluctuation
Ph = lateral production term
pk = vertical production of turbulence energy
Pe = vertical production of dissipation
q = magnitude of the resultant velocity vector (U,V)
72
73
Sx = channel slope in x-direction (-ozb/ox)
t = time
T = mn depth-integrated effective stress tensor per unit mass (Txx' Txy' Tyy)
u* = shear velocity
v. = time average velocity components (u, v, w) 1
v'. =turbulent velocity fluctuations (u', v', w') 1
V = depth-integrated velocity vector (U,V) m w =
0 channel half width
step height
x. = coordinate directions (x,y,z) 1
x = r reattachment length
X = coordinate direction (x,y) m zb = elevation of the channel bottom above an arbitrary datum
z = roughness height 0
z = location of the depth mean velocity above the channel bottom
Greek Symbols
~ = exponent equation (3-11)
o .. = Kronecker delta 1J
~ = denotes finite increment
e = rate of dissipation of turbulence energy per unit mass
e' = fluctuations of the rate of dissipation per unit mass
~ = nondimension local coordinate, y/~s
e = angle that lies between the direction resultant velocity vector, q and the x-axis
K = von Karman's universal constant
74
v = kinematic viscosity of water
vt = effective turbulent eddy viscosity
~ = nondimensional local coordinate, x/~s
p = mass density of water
ok = constant equation (3-19)
ae = constant equation (3-24)
a .. = shear stress tensor per unit mass l.J
I = depth-integrated turbulent stress tensor mn tbm = bottom stress vector
~ = arbitrary scalar function
Subscripts
a = denotes evaluation of functions at the water surface
b = denotes evaluation of functions at the channel bottom
B = bottom of finite control cell
i = tensor notation index
j = tensor notation index
L = left face of finite control cell
m = tensor notation index
n = tensor notation index
r = tensor notation index
R = right face of finite control cell
s = tensor notation index
T = top of finite control cell
w = near wall value
Superscripts
n = time level
= turbulent fluctuation
= depth mean value
75
= depth varing two-dimensional quantity
* = cell face variation
Operations
= time average
= spatial average over control cell
( )R = right cell face average
C\ = left cell face average
( )T = top cell face average
( )B = bottom cell face average
ct_
FLOW ....._ 0 s: I
--...J O"
.... -s:
Figure 1. Two-dimensional definition sketch for a channel expansion.
1-rj ~
OQ
I c:
I 11
CD
I
N
I .
I I t-3
::r
I
t1
h CD
I
CD
I I Ao
I
~ ~
I ::I
I
Cll ~
%1
0 ::I
I Ill
t-
' I
(:l.
CD
I
HI ~
I ::I
~
I rt
x
""" ...
I 0 ::I
I
Cll
I ~
CD
I rt
n
I ::r
I
HI
0 I
t1
I Ill
I
n ::r
I Ill
I
;:I ::I
I (1
) t-'
I (b
I
>< "O
I Ill
;:I
I
Cll """ 0 ::I •
LL
x,IW1
12 I
LEGEND
• Abbott and Kline (1962)
101- • Mohsen et.al. (1967) • Moss et.al. (1977) 0 Kim et.al. (1978) 4 Etheridge and Kemp (1978) • Davis and Snell (1979) 0 Armaly et.al. (1980)
0 81- • Durst et.al. (1980) ... 0
6L ,. .... " • 4
2
o~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
Q01 0.1 1.0 10.0 h/Wo
Figure 3. Experimental measurements of nondimensional reattachment lengths versus the inlet channel aspect ratio.
-....! Cl)
z, w
.s::
.0 N /
x,u
/
I ..,,,,,,. - - - _...:..---
/ / ....
_,_I /
/
Figure 4. Definition sketch for a Cartesian coordinate system with a free surface.
....... \.0
~ .... ()
Q c:: 11
ID
VI • n 0 .§ c:: rt
Q)
rt ..... 0 =' ll) .....
()Q
11 .... Q. •
I--
T t:>
)( 1
t:>
<
• I~
08
•
y
Lx
81
G
Figure 6. Distribution of cell and cell face average quantities.
f -1 .o
82
'fo,1
I I I I I
f ·-- I fo,o - .... - - _, _ - - ·~ - -- 1,0
. I
I I
I I I I I
.:o·~ -- - - _. f1.-1
Figure 7. Nodel information required to compute example right cell face average.
'1>..2 •
-2
u-..-
'1>.., •
-1
I· AS .. 1
r---------- ---, I I I I I I I I
I I •o I
ul <PL • uR•R I I
I I I I I
L- - - - - - - - - - - - --'
0
<P, •
1
Figure 8. Definition sketch for the convective stability analysis.
x AS
CXl l.V
84
fo,o
e fo.-1
Figure 9. Distribution of the normal velocity component grid points about a wall boundary.
85
'fo,1 I I I
t.1.oe---I fo,o _, __ _
- --· f 1,0
Figure 10.
I I
I I I fo-1 ••
Distribution of the streamwise velocity component and water surface elevation grid points about a wall boundary.
y
86
-----, I I ---y I
I I I I I I u---T---'
I I I I I I ~-------'
Figure 11. Computational cells that require modification of the "wall function" boundary specification.
1.0
h
0.85
87
• ·• • • QUICK --- RUNGE-KUTTA
X/tJ.S
Figure 12. Comparison of water surface profile simulations.
. , --+ I) t !' I) I) I) f) f) f) Ill ~ -+ ~ 1' f) f) i'--1'- ~+ ~ y -~~_.-12..~ ~ f) .... f) I) f) I) I) f) £, f) I) I) ~ I) ~ lo+~~ '> L ~ >::. ~) ~ f» ~ ~ r. t> !). i>~~ ~ 1:1 ~ ~ E> v-.+~~ ~ ~ ~ ~" ",""Ill" !I e- ~ +-~+-+-++·'> r " > :;, ~ r> ~ ~ ~ t> ~ t> :> t> t> :> , t> t> ) ~ ti t> ~ ~ ~ I) Ill & & 1> 1> -+-+-++ & & 1> Ill ,._~ ++-t-
~~ v v t> ;. r > ~ ~ ~ t> r· v , ~ t> :.- •· ~ t> ~ ~ .v ~ ~ to ~ I) & f) ,. Ill ~-++ ~ f) ~ Ill Ill & f) ~ f) f) i +-~ x ~ !> !> !) ~ _f, !> !> !> ~ !>_!> !> !> Cl ' !>_!> ~ !'I ti ? ~_\>{>of)~~~ ~f) I) I) f) .. +~-~-+-: ... !I f) !i f) !o...:_+--~·· ~ i.· <t
L!i _q __ 0 1 qlNLET
NOTE: GRID POINTS ARE LOCATED AT THE MIDPOINT OF THE VELOCITY VECTORS
Figure 13. Depth-averaged velocity field for the channel expansion simulation with effective stresses neglected.
00 00
1.0
h
hi NL ET
0.0
/ y x 3.3
AS
Figure 14. Water surface elevations for the channel expansion simulation with effective stresses neglected.
00
'°
90
10
--- k-€ • • • • Experiment
5
0 '--~~~~~-'-~~~~_,;;=-0 0.5
y h
1.0
Figure 15. Comparison of model predictions with the experimental velocity defect measurements of Nakagawa et al. (1975).
91
1.0
k-e • • • • Experiment
y ii 0.5
• 0
0 2 3 4 ..ls.. u2 •
Figure 16. Comparison of model predictions with the experimental turbulence kinetic energy measurements of Nakagawa et al. (1975).
92
40
--- k-€
30 • • • • Experiment
_e_ hU~ 20 •
10
0 ....... ~~~~~ ........ ~~~~~---' 0 0.5 1.0
~
Figure 17. Comparison of model predictions with the experimental energy dissipation rate measurements of Nakagawa et al (1975).
v
L. 100 98 96 96
Figure 18. Water surface elevations in percent of the upstream boundary depth for the standard (k - E) turbulence closure simulation.
98
--<t.
'° w
y
L.
I , _,,, -----""'" ' ' • ' , , ,,,,,.-..-·--------------------------------------1 ' , , ... , , , ,, ,, ,,,,,,,,,,,,._..,~~--------------------------------------\' '--~----........................... _......._.......~ ..................... ___ ~-----------~------------------am-a- 11::ac • •
- - - - - - -ct. q
L__..J qlNLET 0 1
NOTE: GRID POINTS ARE LOCATED AT THE MIDPOINT OF THE VELOCITY VECTORS
Figure 19. Depth-averaged velocity field for the standard (k - t) turbulence closure simulation.
\0 .p..
95
1.5
1.0
0.5 -k-€ • • • Experiment
X-Xr = -2.6 w, o--~~~~--'~~~~~-'-~~~~___J
-0.5 0 ..u.. Uo
0.5 1.0
Figure 20. Comparison of standard (k - E) model predictions with the experimental velocity measurements of Moss et al. (1977).
96
1.5 - k-€ • • • Experiment
1.0
0.5
•
0.5 u Uo
X-Xr w;- = -0.6
1.0
Figure 21. Comparison of standard (k - E) model predictions with the experimental velocity measurements of Moss et al. (1977).
1.5
1.0
0.5
97
-k-€ • • • Experiment
• X-Xr = 1.4 w,
0'--~~~~~~~~~~~......I
0 0.5 u Uo
1.0
Figure 22. Comparison of standard (k - E) model predictions with the experimental velocity measurements of Moss et al. (1977).
I\
Vt I\
Vt <i
2
1
Without Curvature Correction ---- With Curvature Correction
1 I
I
r--
x-x, W1
= 0
,__.. ... --..... I "" -- I ,,."' ........... J
o--~~~~--~~~~~-'-~~~~__,,~~~~~-
o 0.5 1.0 y W1
1.5 2.0
Figure 23. Comparison of predicted eddy viscosities with and without curvature correction at the reattachment point.
\0 00
v
L. t____j 0 1
-- .... ,_ .. I I • - - --
I , , , --------' .... \ • I , ,. --------------------------------------I I , 1 I I I • • • , , "" ;I'// ................. __..._.::.... ___ ~-
I ' • • ' - ..... ....--~~o=;j-------------------------------------~.._.~ ........... ~-... II"';-. ... .., ______ ..... ~~-~~----------------
q Q1NLET
- - - ct. NOTE: GRID POINTS ARE LOCATED AT THE
MIDPOINT OF THE VELOCITY VECTORS
Figure 24. Depth-averaged velocity field for curvature corrected (k - f) turbulence closure simulation.
'° '°
L 1100
"
0 98 94
Figure 25. Water surface elevations in percent of the upstream boundary depth for the curvature corrected (k - f) turbulence closure simulation.
It.
...... 0 0
101
1.5
1.0
0.5 - k-€ • • • Experiment
x-x __ r = -2.6 W1
o.__ _____ _._ _____ __.. _____ __, -0.5 0
u Uo
0.5 1.0
Figure 26. Comparison of curvature corrected (k - t) model predictions with the experimental velocity measurements of Moss et al. (1977).
y w,
102
1.5 -k-€ • • • Experiment
1.0
0.5
x-x ~=-0.6
o ........ __________ L-________ --1
0 0.5 1.0 u Uo
Figure 27. Comparison of curvature corrected (k - t) model predictions with the experimental velocity measurements of Moss et al. (1977).
y w1
103
1.5
- k-€ • • • Experiment
• 1.0
0.5
o---~~~~~-'-~~~~~---'
0 0.5 u Uo
1.0
Figure 28. Comparison of curvature corrected (k - t) model predictions with the experimental velocity measurements of Moss et al. (1977).
104
1.5 ~ • D = 5.0 0 D = 1.0 AO 0 D = 0.2
•o o 1.0
a.a y
W1 oa
0.5 ao •
00.
oo• 0 -0.5 0 0.5 1.0
u UMAX
Figure 29. Comparison of predicted velocity profiles within the recirculation region for varying nondimensional dispersion coefficient.
I\ Vt hU*
105
4
6 6 6 6 6 6
6 3
6
6 D = 5.0 0 D = 1.0
2 0 D = 0.2
1 0 0 0 0 0 0 0 0
0
aP 0 0 0 0 0 0 0 0 0
0 0 0.25 0.5 0.75 1.0
y Wo
Figure 30. Comparison of predicted eddy viscosities at the upstream boundary for varying nondimensional dispersion coefficient.
The vita has been removed from the scanned document
A NUMERICAL SIMULATION OF TWO-DIMENSIONAL
SEPARATED FLOW IN A SYMMETRIC
OPEN-CHANNEL EXPANSION USING THE DEPTH-INTEGRATED TWO-EQUATION
(k-e) TURBULENCE CLOSURE MODEL
by
Raymond Scott Chapman
(ABSTRACT)
Many of the free surf ace flow problems encountered by hydraulic
engineers can be suitably analyzed by means of the depth-integrated equa-
tions of motion. A consequence of adopting a depth-integrated modeling
approach is that closure approximations must be implemented to represent
the so-called effective stresses.
The effective stresses consist of the depth-integrated viscous
stresses, which are usually small and neglected, the depth-integrated
turbulent Reynold's stresses, and additional stresses resulting from the
depth-integration of the nonlinear convective accelerations (here after
called momentum dispersion). Existing closure schemes for momentum dis-
persion lack sufficient numerical and experimental verification to war-
rant consideration at this time, so consequently, attention is focused
on examining closure for the depth-integrated turbulent Reynold's
stresses.
In the present study, an application at the depth-integrated (k-e)
turbulence model is presented for separated flow in a wide, shallow,
rectangular channel with an abrupt expansion in width. The well-known
numerical problems associated with the use of upwind and central finite
differences for convection is overcome by the adoption of the spatially
third-order accurate QUICK finite difference technique. Results pre-
sented show that modification of the depth-integrated (k-e) turbulence
closure model for streamline curvature leads to significant improvement
in the agreement between model predictions and experimental measurements.